Chapitre 6 Couples et suites de variables aléatoires discrètes

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X Y
(Ω,A, P )
(X, Y ) (X, Y )
(X, Y )(Ω) = X(Ω) ×Y(Ω) = {(i, j)i∈X(Ω), j ∈Y(Ω)}
(i, j)∈(X, Y )(Ω) P(X=i)∩(Y=j)
X1X2
PPPPPP
P
X1
X2X1
1
36
1
36
1
36
1
36
1
36
1
36
1
6
1
36
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1
6
1
36
1
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1
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6
1
36
1
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1
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1
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6
1
36
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6
X21
36
1
18
1
12
1
9
5
36
1
6
5
36
1
9
1
12
1
18
1
18
HHHH
H
1
1

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(X, Y )X Y
xi∈X(Ω) P(X=xi)
X=xiY=yj
(Y=yj)yj∈Y(Ω)
Y
P(X=xi) = P
yj∈Y(Ω)
P(X=xi)∩(Y=yj)
yj∈Y(Ω)
P(Y=yj) = P
xi∈X(Ω)
P(X=xi)∩(Y=yj)
(X=xi)X
Y(X=xi)P(X=xi)
P(X=xi)(Y=yj) = P(X=xi)∩(Y=yj)
P(X=xi)
(X=xi)
Y
P(X=xi)∩(Y=yj)=P(X=xi)×P(X=xi)(Y=yj) = . . .
(X=xi) (X=xi)
P(X=xi)
X1X2= 4
xiPX2=4(X1=xi)
1
3
1
3
1
3

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X2(X1= 4)
yj
P(X1=4)(X2=yj)1
6
1
6
1
6
1
6
1
6
1
6
A B
P(A∩B) = P(A)P(B)P(A) = PB(A)P(B) = PA(B)
A B
x∈X(Ω) y∈Y(Ω)
(X=x) (Y=y)
∀x∈X(Ω),∀y∈Y(Ω), P (X=x)∩(Y=y)=P(X=x)P(Y=y)
(X, Y )X Y
XY
cov(X, Y ) = E[X−E(X)] ×[Y−E(Y)] 
cov(X, Y ) = E(XY )−E(X)E(Y)
cov(X, Y ) = EXY −XE(Y)−E(X)Y+E(X)E(Y)=E(XY )−EXE(Y)−EE(X)Y+E(X)E(Y)
=E(XY )−E(X)E(Y)−E(X)E(Y) + E(X)E(Y) = E(XY )−E(X)E(Y)
cov(X, Y ) = cov(Y, X)
Xcov(X, X) = V(X)
(cov(aX +bY, Z) = acov(X, Z) + bcov(Y, Z)
cov(X, aY +bZ) = acov(X, Y ) + bcov(X, Z)

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(X, Y )X Y
(X, Y )
ρ(X, Y ) = cov(X,Y )
σ(X)σ(Y)
X Y
(X, Y )
−1≤ρ(X, Y )≤1
ρ(X, Y )=1 a > 0b∈RY=aX +b
ρ(X, Y ) = −1a < 0b∈RY=aX +b
X Y E(XY ) = E(X)E(Y)
cov(X, Y ) = ρ(X, Y ) = 0
cov(X, Y )6= 0
cov(X, Y ) = 0 X Y
E(X1) = 3,5E(X2) = 7
E(X1X2)
E(X1X2) = P
(x,y)∈(X,Y )(Ω)
xyP (X=x)∩(Y=y)
E(X1X2) = 1
36 1×(2+3+4+5+6+7)+2×(3+4+5+6+7+8)+3×(4+5+6+7+8+9)+4×(5+6+
7 + 8 + 9 + 10) + 5 ×(6 + 7 + 8 + 9 + 10 + 11) + 6 ×(7 + 8 + 9 + 10 + 11 + 12)
E(X1X2) = 1
36 27 + 2 ×33 + 3 ×39 + 4 ×45 + 5 ×51 + 6 ×57
E(X1X2) = 1
36 27 + 66 + 117 + 180 + 255 + 342=210+435+342 6=987 36 =329 12
cov(X1, X2) = 329
12 −7×7 2=329 12 −49×6 2×6=329−294 12 =35 12 6= 0 X1X2
X Y
cov(X1, X2) = cov(X, X +Y) = cov(X, X) + cov(X, Y ) = V(X) + 0 = V(X)
V(X) = 1
6×(1−3,5)2+(2−3,5)2+(3−3,5)2+(4−3,5)2+(5−3,5)2+(6−3,5)2=1 2×25 4+9 4+1 4+1 4+9 4+25 4
=70
24 =35 12

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g(X, Y )
(X, Y )g
(X, Y )(Ω) (X, Y )
g(X, Y )
Z=g(X, Y )
Z(Ω)
z∈Z(Ω) P(Z=z) = P(g(X, Y ) = z)
(X=x)∩(Y=y)
(X, Y )g(X, Y )(Ω) = {g(x, y)x∈X(Ω), y ∈Y(Ω)}
(X=xi)xi∈X(Ω) z∈g(X, Y )(Ω)
Pg(X, Y ) = z=P
xi∈X(Ω)
P(X=xi)∩(g(X, Y ) = z)=P
xi∈X(Ω)
P(X=xi)∩(g(xi, Y ) = z)
g g(xi, Y ) = z Y =....
Y
Pg(X, Y ) = z=P
yj∈Y(Ω)
P(Y=yj)∩(g(X, Y ) = z)=P
yj∈X(Ω)
P(Y=yj)∩(g(X, yj) = z)
(X, Y ) (X+Y)(Ω) = {x+yx∈X(Ω), y ∈Y(Ω)}
z∈(X+Y)(Ω) (X=xi)xi∈X(Ω)
PX+Y=z=P
xi∈X(Ω)
P(X=xi)∩(X+Y=z)=P
xi∈X(Ω)
P(X=xi)∩(Y=z−xi)
(Y=yj)yj∈Y(Ω)
PX+Y=z=P
yj∈Y(Ω)
P(Y=yj)∩(X+Y=z)=P
yj∈Y(Ω)
P(Y=yj)∩(X=z−yj)
(Y=z−xi) (X=z−yj)
z−xiz−yjY(Ω) X(Ω)
(X, Y ) (XY )(Ω) = {xy x∈X(Ω), y ∈Y(Ω)}
z∈(XY )(Ω) (X=xi)xi∈X(Ω)
PXY =z=P
xi∈X(Ω)
P(X=xi)∩(XY =z)=P
xi∈X(Ω)
P(X=xi)∩(Y=z
xi)
(Y=yj)yj∈Y(Ω)
PXY =z=P
yj∈Y(Ω)
P(Y=yj)∩(XY =z)=P
yj∈Y(Ω)
P(Y=yj)∩(X=z
yj)